Download Lecture 2014-12-07

X- and γ -ray spectroscopy
Carlo Ferrigno
Department of astronomy - University of Geneva
https://cms.unige.ch/isdc/ferrigno/
2016 December 7
• Based on Rybick & LightMan Radiative Processes in
astrophysics 1985 Wiley
• Advanced material available in X-ray spectroscopy in
astrophysics, 1997, Paradijs, Jan van, Bleeker, Johan
A.M. (Eds.)
Introduction
Let’s pass from processes giving rise to a continuum radiation to processes producing lines:
• Continuum Spectrum
–
–
–
–
–
Curvature
Compton scattering
Synchrotron
Bremsstrahlung
Pair productiona
• Line spectrum
–
–
–
–
–
–
–
a
Atomic structure
Transition probabilities
Absorption and emission coefficients
Line shapes
Collisional excitation
Photoionization
Conditions of plasma from lines
Feature at 511 keV
Introduction
Emisison line spectrum in X-rays (Hitomi-Collaboration et al.
2016)
Absorption line spectrum in X-rays (Pintore et al. 2016)
1
Principal quantum number
Most of the properties can be derived from classical quantum mechanics whose basic principles
are:
• Action is quantized (Sommerfield):
H
pi qi dqi = ni h
• Heisenberg indetermination principle: states with ∆pi ∆qi < h cannot be distinguished
• Pauli exclusion principle: two Fermions cannot occupy the same cell in phase space. In case
of electrons in atoms, they cannot have the same quantum numbers.
Quantization of action in central potential of atom with central charge Ze leads to the first quantization number n for the energy levels of radial operator:
n2~2
n2
n2
rn =
= a0 ∼ 0.5Å
2
Zme
Z
Z
(1)
where a0 = ~2/me2 ∼ 0.5 Å is the first Bohr radius. Quantization of azimuthal momentum can
be introduced for the second quantum number l in elliptical orbits. Also take into account the finite
nucleus mass by changing electron m into µ = me M/(me + M). A more straightforward way of
considering the angular momentum is the use of Schrodinger equation.
Atomic structure
1
Lines from jump in shells - Order of magnitude
Metalsa
Hydrogen
Line emission or absorption through transition
between orbits with different quantum number n.
Photon energy is:
hν21 = e2Z
1
r1
−
1
r2
2 2
=
Z e
a0
1
n12
−
1
n22
where the constant is know as 1 Rydberg '
13.6 eV.
e2
1 Ry =
2a0
Atomic structure
(2)
Inner shell transitions for heavier atoms are
(3) fluorescence “K", “L” ... lines.
a
astronomical
2
Digression Iron lines
Iron has atomic number 26, the inner shell transitions are in the X-ray domain,
quite isolated from other features and then "easy" to detect. Depending on the
environment, Iron can be ionized at many different level.
Neutral Iron fluorescence Kα is at 6.38 keV, while Fe XVI is at 6.7 keV.
Emission lines for weakly ionized iron. On the
left the Kα lines, on the right, the Kβ lines.
Emission lines for strongly ionized Fe XXIV–
Fe XXVI mostly.
The best-ever X-ray spectrum of a Galaxy cluster.
Atomic structure
3
Schrödinger equation
For an Hamiltonian H , we can write the time-dependent Schrödinger equation
i~
∂ψ
= Hψ
∂t
(4)
And then look for the stationary solutions by separating the time and space parts of the wavefunction ψ , which is possible if H is time independent ψ(t, r) = ψ(r)eiEi /~. We end with the equation:
Hψ = Eψ
(5)
where E is the energy and ψ the eigenfunction of the corresponding energy state. For a nucleus
of charge Ze, neglecting spin, relativistic effects and nuclear effects, the Hamiltonian is




X 1 X e2

 ~2 X
2
−
+
−E 
ψ=0
∇j − Ze

 2m
r
r
j
ij


j
i>j
j
| {z } | {z } | {z }
1
2
(6)
3
where the terms are 1) kinetic, 2) Coulomb, 3) electron self interaction. By expressing length as
function of a0 and energy as e2/a0=27.2 eV, the equation can be written in a dimensionless form


1
2
Atomic structure
X
j
2
∇j + Z
X1
j
X1
−
− E ψ = 0
rj
rij
(7)
i>j
4
Solution of Schrödinger equation in a central field.
The central potential implies that angular momentum is constant and thus the
corresponding operator commutes with the Hamiltonian. It is useful to consider
one electron in a central filed also for high-Z atoms.
Limiting cases:
1
• one electron is far away: he others screen nucleus: V (r ) → Z −N+
,r →∞
r
• one electron is close to the nucleus V (r ) → − Zr , r → 0.
A solution can be found by separating variables and is:
ψ(r , θ, φ) = r −1R(r )Y (θ, φ)
(8)
where the spherical harmonics are defined as
(l − |m|)! 2l + 1
Ylm (θ, φ) =
(l + |m|)! 4π
1/2
|m|
(−1)(l+|m|)/2Pl (cos(θ))eimφ
(9)
where Plm is the associate Legendre function and l and m are integers.
Atomic structure
5
The n, l an m quantum numbers: bound states.
The spherical harmonics are an orthonormal base, solution of each central potential, and are eigenfunctions of the angular momentum operator L = r × p,
expressed in units of ~
L2Ylm (θ, φ) = l(l + 1)Ylm (θ, φ)
Lz Ylm (θ, φ) = mYlm (θ, φ)
(10)
(11)
where l is integer and m = −l...0... + l
The radial part of the eigenfunction is obtained by solving
2
1 d Rnl
l(l + 1)
+ R − V (r ) −
Rnl = 0
2
2
2 dr
2r
(12)
the radial eigenfunction depends on l , but not on m, the index n labels the energy states in increasing order of energy n = l + 1, l + 2, l + 3....
There is also a continuous set of eigenfunctions for unbound states.
Atomic structure
6
Radial functions and orbitals
The bound states are described by
1/2
Z (n − l − 1)!
−p/2 l+1 2l+1
e
p Ln+l (ρ)
Rnl (r ) = −
3
2
n [(n + l)!]
En = −Z 2/2n2(e2/a0)
(13)
(14)
l+1
where ρ = 2Zr /n and L2n+l
(ρ) are the associated Laguerre polynomials.
Normally, the angular momentum quantum number is indicated with letters:
l=0
l=1
l=2
l =3
l= 4
l= 5
l= 6
s
p
d
f
g
h
i
= sharp
= principal
= diffuse
= fundamental
... ...
Atomic structure
7
Spin and many electron systems
In non-relativistic approximation, the spin can be described with a doublet:
1
1 1
0
=
−
=
2
0 2
1
(15)
(relativistic treatment requires the solution of Dirac’s equation).
We have a set of N Fermions, each described by four quantum numbers n, l, m, ms , defining the
orbitals. We need to form the product of the sort
ua(1)ub (2) ... uk (N) ,
(16)
where each subscript a, b...k represents the set of four quantum numbers (n.l, m, ms ), and the
numbers 1...N represent the space and spin coordinates of each electron. These functions include the spacial plus angular parts ψ and the spin.
Particle cannot be distinguished, we make a linear combination of all permutations of the states
and take into account the Pauli exclusion principle. The wave function becomes
(N!)
−1/2
X
p P(ua(1)ub (2) ... uk (N)) ,
(17)
p
where p = ±1, according if the permutation is even or odd. This prevents two electrons being in
the same orbital. There is also complete antisymmetry of the wave function under interchange of
electrons.
Atomic structure
8
Atomic structure
Recap on typical notations:
l = 0 1 2 3 (subshell)
s p d f
n= 0 1 2 3
(shell)
K L M N
Order of filling is the following:
1s , 2s , 2p , 3s , 3p , 4s , 3d , 4p , 5s ,
4d , 5p, 6s , 4f , 5p , 6p , 6s , 6d , ...
Some shells are not in the expected numbering, because for some angular momenta the
shells penetrate closer in the central potential.
Electrons are subject to a quantum mechanics repulsion called exchange potential →
electrons with the same spin are repulsed.
Atomic structure
9
Spectroscopic notation I
• Closed shell are spherically symmetric and thus interactions of these electrons with external fields is minimal
• Non-equivalent electrons differ in either n or l
• Equivalent electrons have same n and l . The notation is:
two equivalent s electrons are: s 2.
two non-equivalent s electrons are: s · s or ss 0.
• Total spin and total angular momentum combine:
S=
P
i
si
L=
P
i li
• In the ground state, Hund’s rules hold
– terms with larger S tend to lie lower in energy, because for the Pauli principle they are further apart.
– among terms with a given spin configuration, those with largest L tend to lie lower in energy because they orbit further apart.
• Due to exchange potential, electrons prefer to occupy the filled shells in
equivalent states. E.g.: 1s 2 2s 2 2 p 2, rather than ... 2p · p .
Atomic structure
10
Spectroscopic notation II
Spectroscopic notation is used to specify how many combinations with different
energy can assume a set of electrons in a given state.
#ms
(Total L)
(18)
• Two non-equivalent electrons in states 1s 2s (excited Helium) can have only
L = 0, but the spin can be S = 0, 1 (1/2-1/2, 1/2+1/2). There can be two possible terms: 1S , for S = 0, called singlet and 3S , for S = 1, called triplet.
• Two equivalent electrons in states s (same shell number) can have only opposite spin: 1S .
• two non-equivalent p electrons, the possible L−S combinations are S = 0, 1,
L = 0, 1, 2, and spectroscopic terms are 1S , 1P , 1D , 3S , 3P , 3D and 1 + 3 +
5 + 3 + 9 + 15 = 36 different states.
• two equivalent p electrons (same shell) can have only 15 distinguishable
states (Exercise).
Atomic structure
11
LS coupling I
Configuration of quantum numbers is not sufficient to determine the state of
ions. For instance the different ml and ms states are all degenerate in a central
potential which does not depend on spin.
Total spin and total angular momentum combine:
S=
P
i
L=
si
P
i li
and then form the total angular momentum
J=S+L
The effect is to split each term into a set of levels, each labelled by the remaining
quantum number J .
The idea is that the central electric field is seen by the moving electron as a
magnetic field. In the rest frame of the electron, we have:
l dU
B=− v×E=
,
(19)
c
mecr dr
e
which interacts with the electron’s magnetic moment µ = − mc
s. And interact as
U = − 12 µ · B.
1
Atomic structure
12
LS coupling II
The spin-orbit Hamiltonian is often written as the sum of interactions of all electrons
X
X
Hso = ξ
Si ·
Li = ξS · L
(20)
i
i
We can find the splitting of a given term as function of the total angular momentum.
J = (S + L) · (S + L) = S2 + L2 + 2S · L
(21)
so that
1
2
2
2
Hso = ξ J − S − L
(22)
2
Since the operators are mutually commuting, the energy shifts are proportional
to J(J + 1)
1
EJ+1 − EJ = C(J + 1)
(23)
2
where C > 0 for shells less than half full (normal) and C < 0 for more than half
full (inverted). Lande interval rule: the spacing between two consecutive levels
of a term is proportional to the larger of the two J values involved.
Atomic structure
13
LS coupling III
The J level of a level is given as a subscript o the term symbol:
2S+1
LJ
(24)
e.g., 3P2, or 2S 1
2
Often the terms of allowed J are expressed as a list: e.g., 2P1/2,3/2. The number
of J values in any term is given by the smaller of (2L + 1) and (2S + 1).
Atomic structure
14
Classical oscillator
• When electrons transit between levels, photons can be absorbed or emitted
at energy
hν = E2 − E1 ,
(25)
where E1 and E2 are the energy associated with the two levels.
• During transition the electron behaves as an oscillator with that resonance
frequency.
• We might perform semi-classical computation, but we will directly compute
the “oscillator strength” derived from quantum mechanics to obtain correct results.
• Line emission m → n is treated using the Einstein A coefficient. The power
emitted per unit volume is:
dP
= Nm hνmn Amn
(26)
dV
where Nm is the number density in level m and hνmn the energy of the transition.
Radiation-Atom interaction
1
Electromagnetic Hamiltonian
Hamiltonian of a particle with charge e in an external EM field
2
2 4 1/2
H = (cp − eA) + m c
+ eφ
(27)
In the non-relativistic limit ignoring the rest mass and using the Coulomb gauge ∇ · A = φ = 0
eA
H=
p−
2m
c
1
2
p2 2
e2A2
+ eφ =
+ eφ
A·p+
2
2m mc
2mc
(28)
if we compute the ration of the terms in A, the linear dominates on the quadratic, unless the phoe
ton density is extremely high (n ∼ 1025 cm−3 )a We will use the term H 1 = (− mc
2 )A · p as a
perturbation to the atomic Hamiltonian.
H = H0 + H1
(29)
where the H 0 is the atomic Hamiltonian, independent of time. We express the atomic eigenvalues and eigenfunction as Ek and φk , so that the zeroth-order time dependent wave functions are
φk exp{−iEk t/~}. The actual wave function can be expanded in this complete set:
ψ(t) =
a
X
ak (t)φk exp{−iEk t/~}
(30)
CMB nγ ∼ 400 cm−3 ; Sun surface: nγ ∼ 1012 cm−3
Radiation-Atom interaction
2
Transition probability
The probability per unit time for a transition from state |ii to state |f i is:

RT 1
1
iωt
1

H
(t)e
dt
H
(ω)
≡

fi
fi
2π 0

2
R ∗ 1 3
4π 1
2
1
wfi =
Hfi (ωfi ) with Hfi (t) ≡ φf H φi d x

~T

ω ≡ Ef −Ei
fi
(31)
~
and the perturbation is active only in the time interval [0, T ]. We assume that
the perturbation is a wave A(r, t) = A(t)eik·r with A(t) limited in time, but large
enough to host many waves; using pj → −i~∇j , we obtain
X
ie~
ik·r
∇j |ii
Hfi (ωfi ) = A(ωfi ) ·
hf |e
mc
1
The transition rate is then
4π 2e2
wfi =
m2c 2T
2
X
2
ik·r
|A(ωfi )| hf |e 1 ·
∇j |ii
(32)
(33)
where the unit vector 1 represents the polarization vector traveling in direction n
ω2
and we can substitute the intensity per unit are and frequency as I = cT
|A(ω)|2.
Radiation-Atom interaction
3
Dipole approximation
Transition probability has terms like
hf |e
ik·r
1·
X
Z
∇j |ii =
φ∗j eik·r1
·
X
∇φj d3x
(34)
2
which we what to expand in k · r as eik·r = 1 + ik · r + 12 (ik · r) and this is appro∆E
priate since k · r ∼ ka0 ∼ a0~c
∼ Z2α << 1
If we retain only the constant term, we get the electric dipole. If this is zero for
symmetry reasons of the electron configurations, we need to climb the higher
terms: magnetic dipole, electric quadrupole, octupole.
For the dipole term, we can write
Z
φ∗j 1
·
X
∇φj d3x = i~−1(1 · pj)fi
(35)
using commutation relations rj p2j − p2j rj = 2i~pj
wfi =
4π 2
~2c
Radiation-Atom interaction
2
|(1 · d)fi | I(ωfi ) with d ≡ e
X
rj
(36)
j
4
Einstein coefficients
A21 is the coefficient of spontaneous emission, B12 is for absorption, and B21 for
stimulated emission. Multiplying the Eistein coefficients to the average flux density, we get the transition probability for each process.
A21 =
3
2hν21
c2
B21
g1B12 = g2B21
(37)
with gi state multiplicity. Let’s relate them to the transition probability.
Radiation-Atom interaction
5
Oscillator strength I
For the case of unpolarized radiation
D
2
|(1 · d)fi |
E
=
1
|dfi |2
3
since hcos θi = 1/3. We can relate the previous computations to the Einstein B
coefficient, after noting that u and l refer to the upper and lower states
hwlu i = Blu Jνlu
We note that Jνul = (4π)−1I(νul ) = 12 I(ωul ), so that
hwlu i =
4π 2
1
2
ul ) =
ul )
|d
|
I(ω
B
I(ω
lu
fi
3c~2
2
By developing the above relations, we can express the Einstein coefficients as (let’s consider nondegenerate states):
2
Blu =
2
Aul =
3
4ωul
|dfi |
Exercise: get the above
Radiation-Atom interaction
3c 3~
'
4c
3 a0
8π 2 |dfi |
= Bul
(38)
α4Z 4 ∼ 109Z 4s−1
(39)
3c~2
6
Oscillator strength II
It is convenient to define the absorption oscillator strength flu by the relations:
4π 2e2
classical
flu ≡ Blu
flu
hνul mc
X
2m
2
flu =
−
E
|d
|
(E
)
u
l
lu
3c~2e2
Blu =
(40)
(41)
The classical B coefficient can be associated to a classical oscillator and expresses the amount of radiation extracted from a beam of radiation.
Z
0
∞
πe2
classical hνul
σ(ν)dν =
= Blu
mc
4π
Order of magnitude for a typical dipole transition rate is Aul ∼ 108 − 1012 s−1.
Which means that exited states have a decay time of the order of the nanosecond or less. Therefore transitions are highly probable.
Radiation-Atom interaction
7
Oscillator strength III
In a system with Z active electrons, the oscillator strength obeys to the rules:
X
n<m
fmn +
X
fmn = Z
(42)
n>m
Known as Reiche-Thomas-Kuhn rule, it states that the total number of equivalent electrons cannot be more than the total number of electrons.
For two levels, following Einstein rules, the oscillator strength has the property
gm fmn = −gn fnm
(43)
where gm,n are the statistical weights. For a level l , g = 2l + 1, for the n-th shell
of a H atom gn = 2n2.
Values of the oscillator strengths are computed via numerical models and also
measured empirically, they are available at the NIST atomic database at https:
//www.nist.gov/pml/atomic-spectra-database
Radiation-Atom interaction
8
Selection rules
In general there is always some probability of transition between states, but
in some cases, it can be exceedingly small. This happens when the transition
probability is strictly zero for dipole, but non null for higher multipoles.
Allowed lines: dipole selection rules:
P
• Laporte’s rule: there are no transitions between states with the same parity (−1)
li
, with li
angular momentum numbers of the individual orbitals.
• Configuration of electrons must change
• Dipole operator is a vector, thus ∆l ± 1 and ∆m = 0, ±1. This rule applies to the moving electron, like in one electron atoms H I He II or the alkali metals
• Rules for multi-electron atoms . They involve the total quantities L,S , and J . A general rule,
which is true also for higher multipoles is that transitions from J = 0 to J = 0 are always forbidden because photon is a boson. We have then in L-S coupling:
∆S = 0
∆L = 0, ±1
∆J = 0, ±1 except J = 0 → J = 0
∆MJ = 0, ±1 except J = 0 → J = 0
Radiation-Atom interaction
(44)
(45)
(46)
(47)
9
Selection rules
Taking the higher terms of the expansion for the transition probability, we get the
magnetic dipole and electric quadrupole.
Forbidden lines: Magnetic dipole (k · r )
• The spontaneous emission coefficient
can be written as Aul =
2
order of the Bohr magneton |µB | =
3
4ωul
3~c 3
|µul |2 where |µul |2 is of the
e~
2me c
• This implies that Aul ∼ 104 s−1 and therefore the state is meta-stable, when Aul > 104 s−1,
the state is semi-forbidden.a
• There are many ways to express the magnetic dipole one is µ = L + 2S, in which we need to
add spin with factor 2 from Dirac’s equation.
• The most important difference from dipole is that parity is unchanged
• No change in electronic configuration is needed
• Spin flip is possible
a
Exercise: derive the rate from eq. (38) using the expansion in k · r, which is the suppression factor?
Radiation-Atom interaction
10
Selection rules
Taking the higher terms of the expansion for the transition probability, we get the
magnetic dipole and electric quadrupole.
Forbidden lines: electric quadrupole E2 (k · r )
• No change in parity
• Term is quadratic in r: at least one change in nl
∆S = 0
∆L = 0, ±1, ±2
∆J = 0, ±1, ±2 except J = 0 → J = 0
(48)
(49)
(50)
Two photon transitions
• Transitions with J = 0 ↔ 0 are not allowed for nature of photons (1s2 − 1s2s 1S0)
• In H-like atoms transitions are not allowed L = 0 ↔ 0 (1s − 2s)
• Only emission of two photons can preserve angular momentum
• Conservation of energy requires that the sum of energies is preserved
• The most probable case is the emission of two dipole photons within the allowed time
∆t∆E ∼ ~a
a
Exercise: using the probability of dipole and the uncertainty principle, derive the suppression factor of
the emission coefficient from the dipole case.
Radiation-Atom interaction
11
Line width I - natural broadening
• Uncertainty principle implies natural broadening of lines ∆E∆t ∼ ~ . The
P
0
0 , where n
spontaneous decay from a state n proceeds at rate γ =
A
0
nn
n
are the states with lower energies.
• If radiation is present, we should add the induced rate
• The coefficient of the wave function for the perturbed Hamiltonian (eq. 30) is
thus of the form e−γt/2 and leads to a decay of the electric field of the same
factor (energy as e−γt )
• The Fourier transform of a decaying sinusoid is a Lorentzian function
γ/4π 2
φ(ν) =
(51)
2
2
(ν − ν0) + (γ/4π)
where ν0 is the transition frequency and γ is the line width (Full width half
maximum)
• The above is valid strictly only for the ground state, since it has infinite live
time, if both states are broadened (finite live time), γ = γl + γu .
Radiation-Atom interaction
12
Line width II - Doppler broadening
• Astronomical systems are hot: atoms are shacked and emit in a frame which is not the observer’s frame, here ν0 is the rest-frame frequency.
ν0 v z
ν − ν0 =
(52)
c
• The number of atoms with the
velocity
in the range vz − vz + dz is proportional to the
2
m v 2
Maxwellian distribution exp − 2akTz dvz , with ma mass of the atom
• By substituting eq. (52), we obtain the line profile, in the assumption that natural broadening is
negligible
φ(ν) =
1
2
2
√ e−(ν−ν0) /(∆νD )
∆νD π
(53)
where the Doppler width is defined as
∆νD =
ν0
c
s
2kT
ma
• If microturbulence is present, stochastic motion << mean free path of atoms
s
ν0 2kT
∆νD =
+ξ
c
ma
with ξ rms measure of the turbulent velocity.
Radiation-Atom interaction
13
Complication on Line
• Fine structure lines are forbidden lines with ground state multiplet, usually
seen in IR and cool gas
• Hyperfine structure lines interaction with nuclear spin, e.g. electron spin flip
produces the 21 cm line of neutral Hydrogen
• Satellite lines Lines produced from excited atoms, in which inner shell electrons are missing
– perturbed energy levels
– slightly different energy levels from the lines of the ground state
– They appear as satellite of the lines in the ground state
Radiation-Atom interaction
14
What next
In the next lecture, we will apply these basic notions to the explain some
features in high-resolution X-ray spectra and derive the physical conditions of
the emitting/absorbing regions.
Summary
1
Summary
Bibliography
0–30a
[1607.07420v1]
Pintore, F., Sanna, A., Di Salvo, T., et al. 2016, arXiv.org [1601.05215v1]
Hitomi-Collaboration, Aharonian, F. A., Akamatsu, H., et al. 2016, arXiv.org
0